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practice(exam2)MA262fall2011

practice(exam2)MA262fall2011 - PRACTICE PROBLEMS FOR EXAM 2...

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PRACTICE PROBLEMS FOR EXAM 2 - MA 262 FALL 2011 INSTRUCTOR: RAPHAEL HORA 1. If A = 1 1 1 1 3 - 4 1 2 - 5 , then det( - 3 A ) =? (Hint: If A is an n x n matrix, then det( cA ) = c n det( A ), for any c R . ) 2. Compute 5 5 5 - 2 - 4 10 1 3 - 4 . 3. If A, B and C are 5 x 5 matrices such that det( A ) = 5, det( B ) = - 2 and det( C ) = - 4, then det( - AB - 1 C T )=? 4. Determine all values of k such that the vectors (1 , 1 , 1) , (2 , k, 3) and ( - 1 , - 1 , k - 2) are a basis for R 3 . (Hint: Find all values of k such that the determinant of the matrix whose rows are the given vectors is different from zero.) 5. Which of the following subsets S of the given vector space V is a subspace? (1) V = C 2 ( R ), S = { y 00 - (ln x + cos 3 x ) y 0 + (1 + e - 2 x sin x ) y = 0 } ; (2) V = P 3 , S = { x 3 + ax 2 + bx : a, b R } ; (3) V = R 3 , S = { ( x + 2 y, x - y, 0) : x, y R } ; (4) V = R 2 , S = { ( a, b ) : a 0 , b R } ; (5) V = P 4 , S consists of all polynomials with degree 2 and the zero polynomial; (6) V = M 3 ( R ) , S = { A M 3 ( R ) : det( A ) = 0 } ; (7) V = C 1 ( R ), S = { f V : f 0 (0) = f 0 (1) } . 6. Find a such that the vector (1,1,1) can be written as a linear combination of (2,-1,1) and (3 a ,3,-1).
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